When expressing, did you turn to the lowest power? If the power is too high, your poor nipples will suffer. Also, have to make sure that the funnel is not too tight; there should be some space for nipple to be pulled in.
For relief, you can try nipple creams by mendela and lansinoh. Remember not to wash your nipples with soap or they'll be even dryer and more sore.
Why dont't you try latching baby on directly? baby's the best pump. Some newborn bbs are very sleepy and need to be woken up for feeds. Try not to let more than 3-4 hours pass between feeds.
Your breasts are empty when you squeeze the aerole and there's no milk. Try not to express too much as this may stimulate the production of more milk, and cause even more engorgement.